DLMF:31.10.E11 (Q9062)

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DLMF:31.10.E11
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    Statements

    Defining formula
    𝒦 ⁒ ( z , t ) = ( z ⁒ t - a ) 1 2 - Ξ΄ - Οƒ ⁒ ( z ⁒ t / a ) - 1 2 + Ξ΄ + Οƒ - Ξ± ⁒ F 1 2 ⁑ ( 1 2 - Ξ΄ - Οƒ + Ξ± , 3 2 - Ξ΄ - Οƒ + Ξ± - Ξ³ Ξ± - Ξ² + 1 ; a z ⁒ t ) ⁒ P ⁑ { 0 1 ∞ 0 0 - 1 2 + Ξ΄ + Οƒ ( z - a ) ⁒ ( t - a ) ( 1 - a ) ⁒ ( z ⁒ t - a ) 1 - Ο΅ 1 - Ξ΄ - 1 2 + Ο΅ - Οƒ } . 𝒦 𝑧 𝑑 superscript 𝑧 𝑑 π‘Ž 1 2 𝛿 𝜎 superscript 𝑧 𝑑 π‘Ž 1 2 𝛿 𝜎 𝛼 Gauss-hypergeometric-F-as-2F1 1 2 𝛿 𝜎 𝛼 3 2 𝛿 𝜎 𝛼 𝛾 𝛼 𝛽 1 π‘Ž 𝑧 𝑑 Riemann-P-symbol 0 1 absent 0 0 1 2 𝛿 𝜎 𝑧 π‘Ž 𝑑 π‘Ž 1 π‘Ž 𝑧 𝑑 π‘Ž 1 italic-Ο΅ 1 𝛿 1 2 italic-Ο΅ 𝜎 absent {\displaystyle{\displaystyle\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma% }\left(\ifrac{zt}{a}\right)^{-\frac{1}{2}+\delta+\sigma-\alpha}\*{{}_{2}F_{1}}% \left({\frac{1}{2}-\delta-\sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-% \gamma\atop\alpha-\beta+1};\frac{a}{zt}\right)\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.}}
    0 references
    DLMF id
    DLMF:31.10.E11
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁑ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5aadec
    0 references
    Riemann’s $$P$$ -symbol for solutions of the generalized hypergeometric differential equation
    Defining formula
    P ⁑ { Ξ± Ξ² Ξ³ a 1 b 1 c 1 z a 2 b 2 c 2 } Riemann-P-symbol 𝛼 𝛽 𝛾 absent subscript π‘Ž 1 subscript 𝑏 1 subscript 𝑐 1 𝑧 subscript π‘Ž 2 subscript 𝑏 2 subscript 𝑐 2 absent {\displaystyle{\displaystyle P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}}}
    xml-id
    C15.S11.E3.m2aadec
    0 references
    Q12457
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C31.S1.XMD1.m1edec
    0 references
    Q12458
    Defining formula
    Ξ³ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C31.S1.XMD10.m1fdec
    0 references
    Q12459
    Defining formula
    Ξ΄ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C31.S1.XMD11.m1fdec
    0 references
    Q12460
    Defining formula
    Ο΅ italic-Ο΅ {\displaystyle{\displaystyle\epsilon}}
    xml-id
    C31.S1.XMD12.m1fdec
    0 references
    Q12461
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C31.S1.XMD6.m1edec
    0 references
    Q12463
    Defining formula
    Ξ± 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C31.S1.XMD8.m1ddec
    0 references
    Q12464
    Defining formula
    Ξ² 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C31.S1.XMD9.m1ddec
    0 references
    kernel (locally)
    Defining formula
    𝒦 ⁒ ( z , t ) 𝒦 𝑧 𝑑 {\displaystyle{\displaystyle\mathcal{K}(z,t)}}
    xml-id
    C31.S10.XMD4.m1fdec
    0 references
    Q12497
    Defining formula
    Οƒ 𝜎 {\displaystyle{\displaystyle\sigma}}
    xml-id
    C31.S10.XMD8.m1bdec
    0 references
     
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